Parallel circuits - Overall resistance decreases with additional resistor

Question

Let's say that there is a parallel circuit with two identical resistors in parallel with each other. If a third resistor, identical to the other two, is added in parallel with the first two, the overall resistance decreases.

Why does this overall resistance decrease?

Answer 1

For resistors $R_1, R_2, \dots, R_N$ in parallel, the equivalent resistance $R_e$ is given by $$ \frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_N} $$ If two resistors with equal resistance $R_1 = R_2 = R$ are in parallel, then this gives $$ R_e^{(2)} = \frac{R_1R_2}{R_1+R_2} = \frac{R^2}{2R} = \frac{R}{2} $$ If three identical resistors $R_1 = R_2 = R_3 = R$ are in parallel, then the equivalent resistance is $$ R_e^{(3)} = \frac{R_1R_2R_3}{R_1R_2 + R_2R_3 + R_3R_1} = \frac{R^3}{3R^2} = \frac{R}{3} $$ In fact, for $N$ identical resistors one has $$ \frac{1}{R_3^{(N)}} = \frac{N}{R} $$ so that $$ R_e^{(N)} = \frac{R}{N} $$ and therefore the resistance decreases with the addition of each successive resistor in parallel.

Answer 2

Why does this overall resistance decrease?

A more elegant, sophisticated way to see why is through the notion of duality.

In electric circuit theory, conductance (the reciprocal of resistance) is dual to resistance. Other dual pairs are:

voltage - current

series - parallel

inductance - capacitance

Thevenin - Norton

and so on ...

For example, consider Ohm's Law: $v = iR$. The dual is: $i = vG$

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You probably intuitively understand that adding a resistor in series increases the total resistance.

The dual of this is adding a conductance in parallel increases the total conductance.

But, if the conductance increases, the reciprocal, i.e., the resistance, decreases.

Mathematically:

Conductances in parallel add:

$G_{total}=G_1 + G_2 + G_3 = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} = \dfrac{1}{R_{total}}$

or

$R_{total} = \dfrac{1}{\dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}}$

Now, it's clear that adding another resistor in parallel, increases the denominator thus decreasing the total resistance.

Answer 3

The reason adding resistors in parallel decreases overall resistance is that adding resistors in parallel increases the effective cross sectional area of the circuit. Current is proportional to cross sectional area, so the overall current drawn from the battery (assuming a DC circuit) increases. However, adding a resistor in parallel doesn't change the potential difference across the new resistor or the other existing parallel resistors (that follows from the very definition of parallel). If the circuit obeys the macroscopic Ohm's law ($\Delta V = IR$), then resistance is the quotient of potential difference and current. Potential difference doesn't change when adding a new parallel resistor, but current increases so the quotient decreases. The accepted answer is mathematically correct, but doesn't actually address your question.

Answer 4

Suppose we have a simple circuit with only one resistor. Assume the voltage across the circuit is 10V, and current is say, 2Amps. Now assume we add an additional component to the circuit in parallel as in the following diagram:

The voltage accross each component is not reduced by this action, and now we have a 10V pull over both components in the circuit.

If we consider the first component, we can use the formula $V = I_1R_1$, to find the current to be $I_1 = \frac{V}{R_1}$ (This will equal 2Amps, juas as in the original circuit). Similarly, we can find the current over the second component using the formula $V=I_2R_2$ and therefore $I_2 = \frac{V}{R_2}$. The voltage here is the same as before, because adding additional resistors in parallel does not reduce the pulling power of the battery.

The total current in the circuit is then obtained by adding the current flowing in each of the components, to give $I_T = I_1 + I_2 = \frac{V}{R_1} + \frac{V}{R_2} = V(\frac{1}{R_1} + \frac{1}{R_2})$.

Now we can think of the two resistors as one big resistor, of resistance $R_T$, and use the formula $V = I_TR_T$.

Therefore, $R_T = \frac{V}{V(\frac{1}{R_1} + \frac{1}{R_2})} = \frac{1}{(\frac{1}{R_1} + \frac{1}{R_2})}$, which is the formula used in one of the above answers.

We now can see intuitively why the resistance has decreased. This is because adding the second resistor allowed for an additional current $I_2$, which combines with the original current, $I_1$, to form a larger current than before. When current increases over a constant voltage, we can say that the total resistance of the circuit has decreased.

Answer 5

I almost didn't answer, but maybe it can help someone better understand the accepted answer.

Some people will cringe at the thought of this, but I often find myself comparing electricity to plumbing. It works reasonably well in your question and helps me visualize joshphysics' correct answer.

It is the same as a large tub filled with holes. Adding more drain pipes will cause the tub to drain faster. In this case, the drain pipes are more resistors. Adding a tiny drain hole (representing an additional resistor of large value) will undeniably result in more current flow.

Answer 6

The basic idea is that all resistors can be modeled as a single material which has a resistance that is a function of its cross-sectional area $A$ only. (To be precise this is because all resistors have $0<R<\infty$, and for any resistor, $R$ is proportional to $L/A$ [this is the ONLY assumption needed for this derivation], so for fixed $L$ there always exists such an $A$ such that $0<1/A<\infty$ allowing us to make any resistance we like.)

The same can be done for the lengths -- fix the area, and adjust the length to make any resistance you like.

First, the series circuit. Let us consider, without loss of generality, two resistors $R_1$ and $R_2$ of different lengths and all other things equal. When we smush them together the length increases, and $L=L_1+L_2$, and $R$ is proportional to $L$, so the resistance of the "equivalent resistor" $R_{eq}=R_1+R_2.$

Now imagine a basic parallel section of a circuit with two resistors in parallel, again $R_1$ and $R_2$. A current $I_1$ and $I_2$ flows into each one with the total current being equal to $I$. Now consider a different, but equivalent picture. It's a little tough to imagine but the picture is simple enough to draw -- it's taking one parallel circuit flow and turning it into two coaxial current flows.

Consider a new equivalent resistor which is $R_1$ with an $R_2$ "tube" wrapped around its exterior (so two coaxial cylinders of the same material.) The cylindrical section of $R_2$ wrapped around $R_1$ has the original cross sectional area as the cylindrical version of $R_2$. The total area of the new resistor is now $ A=A_1+A_2.$

The two wires which carry currents $I_1$ and $I_2$ are now coaxial, with current $I_1$ (say) flowing on the interior and $I_2$ flowing on the exterior. Let the cross sectional area of wire $I_1$ (which is arbitrary) be equal to the area of $R_1$, and the same for $I_2$. Now it is clear that all we have done is make the area of our resistor larger. The total current flowing through it, $I_1+I_2=I$, is unchanged.

Since the resistance goes like the inverse of the area, $R$ goes like $1/A$, so $A_1$ goes like $1/R_1$, etc. From above, we get the rule that $1/R_{eq}=1/R_1+1/R_2$.

In short, a series resistor is like a longer resistor and a parallel resistor is like a fatter one. Then it is easy to see why series resistors increase the overall resistance and parallel resistors decrease it. This also generalizes to any number of resistors in series or parallel.